Khasminskii-type Condition Research Articles

Convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching

In this paper, we concern the strong convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching (SPDEs-MS). We present the balanced Euler method of SPDEs-MS and consider its moment boundedness under polynomial growth condition plus Khasminskii-type condition. We also study its strong convergence order and its mean-square stability. Two numerical examples are given to illustrate the theoretical results.

Convergence Rate of the Diffused Split-Step Truncated Euler–Maruyama Method for Stochastic Pantograph Models with Lévy Leaps

This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in Lp(p≥2) sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in Lp(0<p<2) sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples.

Delay Ait-Sahalia-type interest rate model with jumps and its strong approximation

Abstract In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments.

The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero

Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.

Exponential stability of θ-EM method for nonlinear stochastic Volterra integro-differential equations

Mean square exponential stability of both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For 12<θ≤1, we prove that both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If 0≤θ≤12, θ-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on f. By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that θ-EM method is almost surely exponentially stable. An example is provided to support our conclusions.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps.

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay

The main goal of this paper is to establish the Lq-convergence of the truncated Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay under the Khasminskii-type condition. Whole consideration is influenced by the presence of the neutral term and delay function. The main theoretical result is illustrated by an example. Since the main result is related to the Lq-convergence of the Euler-Maruyama method under the global Lipschitz condition on the coefficients of the equation under consideration, for completeness of the paper, the appropriate results are given in Appendix.

Numerical method of highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching

In this paper, we establish a partially truncated Euler–Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.

Truncated EM numerical method for generalised Ait-Sahalia-type interest rate model with delay

The original Ait-Sahalia model of the spot interest rate proposed by Ait-Sahalia assumes constant volatility. As supported by several empirical studies, volatility is never constant in most financial markets. From application viewpoint, it is important we generalise the Ait-Sahalia model to incorporate volatility as a function of delay in the spot rate. In this paper, we study analytical properties for the true solution of this model and construct several new techniques of the truncated Euler–Maruyama (EM) method to study properties of the numerical solutions under the local Lipschitz condition plus Khasminskii-type condition. Finally, we justify that the truncated EM approximate solution can be used within a Monte Carlo scheme for numerical valuations of some financial instruments such as options and bonds.

The truncated Euler–Maruyama method for stochastic differential equations with piecewise continuous arguments driven by Lévy noise

ABSTRACT This paper aims to consider stochastic differential equations with piecewise continuous arguments (SDEPCAs) driven by Lévy noise where both drift and diffusion coefficients satisfy local Lipschitz condition plus Khasminskii-type condition and the jump coefficient grows linearly. We present the explicit truncated Euler–Maruyama method. We study its moment boundedness and its strong convergence. Moreover, the convergence rate is shown to be close to that of the classical Euler method under additional conditions.

Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Guo et al. [GMY17] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations(SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [GMY17] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we consider the theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The existence, uniqueness and pth moment boundedness of the analytic solutions are investigated. Euler method is shown to be divergent in the strong mean square sense for super linear growth coefficients, so the truncated Euler-Maruyama method is presented and its moment boundedness and Lq-convergence are shown. Moreover, its pth moment boundedness and Lq-convergence (q∈[2,p) and p is a parameter in Khasminskii-type condition) rate are given under Local Lipschitz condition and Khasminskii-type condition. The theoretical results are illustrated by some numerical examples.

Convergence rate of the truncated Milstein method of stochastic differential delay equations

This paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations (SDDEs) without the linear growth condition. On the one hand, these nonlinear SDDEs do not have explicit solutions, therefore implementable numerical methods for such SDDEs are required. On the other hand, the implicit Euler methods are known to converge strongly to the exact solution of such SDDEs. However, they require additional computational efforts. In this article, we propose the truncated Milstein method which is an explicit method under the local Lipschitz condition plus Khasminskii-type condition, study its pth monent boundedness (p is a parameter in Khasminskii-type condition) and show that its rate of strong convergence is close to one.

The Partially Truncated Euler–Maruyama Method for Highly Nonlinear Stochastic Delay Differential Equations with Markovian Switching

A class of super-linear stochastic delay differential equations (SDDEs) with variable delay and Markovian switching is considered. The main aim of this paper is to investigate the convergence and stability properties of partially truncated Euler–Maruyama (EM) method applied to the SDDEs with variable delay and Markovian switching under the generalized Khasminskii-type condition.

The truncated EM method for stochastic differential equations with Poisson jumps

In this paper, we use the truncated Euler–Maruyama (EM) method to study the finite time strong convergence for SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time L r ( r ≥ 2 ) -convergence order when the drift and diffusion coefficients satisfy the super-linear growth condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal L r - convergence order is close to 1. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the L r ( 0 < r < 2 ) -convergence results are also investigated and the optimal L r - convergence order is shown to be not greater than 1 ∕ 4 . Moreover, we prove that the truncated EM method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying SDEs with Poisson jumps. Several examples are given to illustrate our results.

Convergence rate and stability of the truncated Euler–Maruyama method for stochastic differential equations

Abstract Recently, Mao (2015) developed a new explicit method, called the truncated Euler–Maruyama (EM) method, for the nonlinear SDE and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. In his another follow-up paper (Mao, 2016), he discussed the rates of L q -convergence of the truncated EM method for q ≥ 2 and showed that the order of L q -convergence can be arbitrarily close to q ∕ 2 under some additional conditions. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to establish the convergence rate without these restrictions. The other aim is to study the stability of the truncated EM method. The advantages of our new results will be highlighted by the comparisons with the results in Mao (2015, 2016) as well as others on the tamed EM and implicit methods.

Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations

This paper establishes the convergence of a class of highly nonlinear stochastic differential delay equations without the linear growth condition replacing by Khasminskii-type condition, so the convergence criteria here may cover a wider class of nonlinear systems. Our aim is to propose the partially truncated Euler–Maruyama method for stochastic differential delay equations d y ( t ) = f ( y ( t ) , y ( t − τ ) ) d t + g ( y ( t ) , y ( t − τ ) ) d w ( t ) and consider the strong- L q convergence for 2 ≤ q < p under the local Lipschitz condition plus Khasminskii-type condition, and p is a parameter in Khasminskii-type condition.

Strong convergence of the truncated Euler–Maruyama method for stochastic functional differential equations

ABSTRACTIn this paper, we establish the truncated Euler–Maruyama (EM) method for stochastic functional differential equation (SFDE) and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong- convergence for , and p is a parameter in Khasminskii-type condition. We also discussed the rates of -convergence for the truncated EM method.

Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations

The sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered, with a Khasminskii-type condition. The almost sure exponential stability of the numerical solutions by the Euler–Maruyama method and the backward Euler–Maruyama method is also discussed, based on the discrete semimartingale convergence theorem. We present the sufficient conditions for the stability of the Euler–Maruyama method, with one extra condition when compared with the exact solution. We show that the backward Euler–Maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution.

The truncated Euler\u2013Maruyama method for stochastic differential delay equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in Lp) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.